# Am I wrong or is my professor wrong? (Basic measure theory/set theory)

Edit:

Here is an image of the question:

Edit 2:

In my professor's e-mail reply to me, my professor wrote that, among other things, "The polynomials of degree $$N$$ certainly include all the monomials of degree $$N$$ or less." Isn't this totally wrong? A polynomial of degree 5 must have nonzero coeffcient $$a_5$$, so the set of all polynomials of degree 5 cannot include $$x^4$$ because $$x^4$$ has coefficent zero for the $$x^5$$ term.

Is this Twilight Zone stuff or am I missing something totally obvious?

In a measure theory course I was given the following problem.

Question:

Define $$A_i = \{ a_i x^i \colon a_i \in \mathbb R\}$$ for $$i = 0, 1, 2, \dots$$ . Construct an increasing sequence of sets that produces $$\bigcup_{i = 0}^{\infty} A_i$$.

My Answer:

To construct the desired sequence of sets, make the definitions

\begin{align*} B_k &:= \bigcup_{j = 0}^k A_j \text{ for } k = 0, 1, 2, \dots \ . \end{align*}

This is an increasing sequence of sets because $$\bigcup_{j = 0}^k A_j \subseteq \bigcup_{j = 0}^l A_j$$ if $$k < l$$.

I assume that the word produces" means that we are expected to prove that $$\bigcup_{j = 0}^{\infty} B_j = \bigcup_{j = 0}^{\infty} A_j$$. First consider $$x \in \bigcup_{j = 0}^{\infty} B_j$$. Then there is some $$k \in \{ 0, 1, 2, \dots \}$$ such that $$x \in B_k$$, which by definition means that there is some $$l \in \{ 0, 1, 2, \dots, k \}$$ such that $$x \in A_l$$. Therefore $$\bigcup_{j = 0}^{\infty} B_j \subseteq \bigcup_{j = 0}^{\infty} A_j$$.

Conversely, consider $$x \in \bigcup_{j = 0}^{\infty} A_j$$. There is some $$k \in \{ 0, 1, 2, \dots \}$$ such that $$x \in A_k$$, which certainly implies that $$x \in B_k$$. Therefore $$\bigcup_{j = 0}^{\infty} A_j \subseteq \bigcup_{j = 0}^{\infty} B_j$$.

The two paragraphs above prove that

\begin{align*} \bigcup_{j = 0}^{\infty} B_j &= \bigcup_{j = 0}^{\infty} A_j. \end{align*}

Commentary:

I was given zero marks for this answer. My professor's feedback was as shown below (this is verbatim; he didn't use LaTeX).

what does this formal setting mean.

think of a0 a0 + a1x a0 + a1x + a2x^2 ...

I don't get this at all. He defines $$A_i$$ to be the set of all monomials of degree $$i$$, unioned with zero. Therefore the union of all $$A_i$$ is all monomials of any degree. I defined $$B_k$$ to be the set of all monomials of degree at most $$k$$. Clearly the $$B_k$$ form an increasing sequence and their union is the set of all monomials of any degree.

Why is he referring to polynomials in his feedback? I don't get it. Nothing in his definition of $$A_i$$ permits general polynomials (addition of monomials). For example, $$3x + 1 \notin \bigcup_{i = 0}^{\infty} A_i$$, although $$1 \in A_0$$ and $$3x \in A_1$$.

I appreciate any feedback because I'm stumped.

• Comments are not for extended discussion; this conversation has been moved to chat. Mar 1 at 12:37
• I don't think that this is a good question for Math SE. It appears (to me) that there is some kind of miscommunication between you and your professor. We cannot read your professor's mind, thus it seems that you need to contact them to get your questions answered. Mar 1 at 12:39
• "Who's right, me or my prof" don't make good titles. Try instead, to pose the question, and ask for solution verification, with no mention of your professor. Mar 1 at 14:37

## 2 Answers

I believe you are correct and your professor is in error. To that end, I would suggest contacting him regarding this problem and explaining as clearly as you can why you believe your interpretation is correct. Your professor is a professional in the mathematics world, they should have enough lived experience with mathematics to appreciate that they are only human and will make errors sometimes (and should generally have the humility to admit it once they recognize it as an error). To that end I will suggest the following:

1. Be respectful and professional in your tone.
2. Explain clearly why you believe your interpretation is correct, and his, in error (I would strongly encourage you to point out how the problem's definition of $$A_i$$ works for some small examples (like $$i=0$$, $$1$$, and $$2$$). I would even go so far as to show why I would object to the idea of any expression of the form $$a_1 x+a_0$$ (for nonzero $$a_1$$ and $$a_0$$) being in the union $$A_0 \cup A_1=\{a_i x^i | i=0,1;a_i\in \mathbb{R}\}$$.
• Thank you. I have sent an e-mail to the professor and tried to follow your advice. I await the professor's reply. Mar 1 at 3:45

I think the professor intends for $$a_ix^i$$ to be interpreted as the sum $$\sum_{n=0}^i a_nx^n$$. This is poor use of Einstein summation notation because labelling the set $$A_i$$ suggests that $$i$$ is fixed, while the $$a_ix^i$$ in the definition of the set uses $$i$$ as a dummy index. This would explain their response referring to polynomials.

Regarding Edit 2, I agree with OP that the polynomials of degree $$N$$ do not include the polynomials of degree less than $$N$$. Their span would, though.

• This may be the confusion of the professor at some level, but it still does not explain why the answer that uses the sets $B_i$ got zero points. Mar 1 at 11:17